1. Field of the Invention
This invention generally relates to optical communications and, more particularly, to a transmitter filter that pre-compensates for the effects of chromatic dispersion in an optical communications channel.
2. Description of the Related Art
Wikipedia notes that in optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency. Media having such a property are termed dispersive media. Dispersion is sometimes called chromatic dispersion to emphasize its wavelength-dependent nature, or group-velocity dispersion (GVD) to emphasize the role of the group velocity.
The most familiar example of dispersion is probably a rainbow, in which dispersion causes the spatial separation of a white light into components of different wavelengths (different colors). However, dispersion also has an effect in many other circumstances: for example, GVD causes pulses to spread in optical fibers, degrading signals over long distances; also, a cancellation between group-velocity dispersion and nonlinear effects leads to soliton waves.
There are generally two sources of dispersion: material dispersion and waveguide dispersion. Material dispersion comes from a frequency-dependent response of a material to waves. For example, material dispersion leads to undesired chromatic aberration in a lens or the separation of colors in a prism. Waveguide dispersion occurs when the speed of a wave in a waveguide (such as an optical fiber) depends on its frequency for geometric reasons, independent of any frequency dependence of the materials from which it is constructed. More generally, “waveguide” dispersion can occur for waves propagating through any inhomogeneous structure (e.g., a photonic crystal), whether or not the waves are confined to some region. In general, both types of dispersion may be present, although they are not strictly additive. Their combination leads to signal degradation in optical fibers for telecommunications, because the varying delay in arrival time between different components of a signal “smears out” the signal in time.
The phase velocity, v, of a wave in a given uniform medium is given by
  v  =      c    n  
where c is the speed of light in a vacuum and n is the refractive index of the medium.
In general, the refractive index is some function of the frequency f of the light, thus n=n(f), or alternatively, with respect to the wave's wavelength n=n(λ). The wavelength dependence of a material's refractive index is usually quantified by its Abbe number or its coefficients in an empirical formula such as the Cauchy or Sellmeier equations.
Because of the Kramers-Kronig relations, the wavelength dependence of the real part of the refractive index is related to the material absorption, described by the imaginary part of the refractive index (also called the extinction coefficient). In particular, for non-magnetic materials (μ=μ0), the susceptibility x that appears in the Kramers-Kronig relations is the electric susceptibility Xe=n2−1.
Since that refractive index varies with wavelength, it follows that the angle by which the light is refracted will also vary with wavelength, causing an angular separation of the colors known as angular dispersion.
For visible light, refraction indices n of most transparent materials (e.g., air, glasses) decrease with increasing wavelength λ:
1<n(λred)<n(λyellow)<n(λblue)
or alternatively:
            ⅆ      n              ⅆ      λ        <  0.
In this case, the medium is said to have normal dispersion. Whereas, if the index increases with increasing wavelength (which is typically the case for X-rays), the medium is said to have anomalous dispersion.
At the interface of such a material with air or vacuum (index of ˜1), Snell's law predicts that light incident at an angle θ to the normal will be refracted at an angle arcsin(sin(θ)/n). Thus, blue light, with a higher refractive index, will be bent more strongly than red light, resulting in the well-known rainbow pattern.
Another consequence of dispersion manifests itself as a temporal effect. The formula v=c/n calculates the phase velocity of a wave; this is the velocity at which the phase of any one frequency component of the wave will propagate. This is not the same as the group velocity of the wave, which is the rate at which changes in amplitude (known as the envelope of the wave) will propagate. For a homogeneous medium, the group velocity vg is related to the phase velocity by (here λ is the wavelength in vacuum, not in the medium):
      v    g    =                    c        ⁡                  (                      n            -                          λ              ⁢                                                ⅆ                  n                                                  ⅆ                  λ                                                              )                            -        1              .  
The group velocity vg is often thought of as the velocity at which energy or information is conveyed along the wave. In most cases this is true, and the group velocity can be thought of as the signal velocity of the waveform. In some unusual circumstances, called cases of anomalous dispersion, the rate of change of the index of refraction with respect to the wavelength changes sign, in which case it is possible for the group velocity to exceed the speed of light (vg>c). Anomalous dispersion occurs, for instance, where the wavelength of the light is close to an absorption resonance of the medium. When the dispersion is anomalous, however, group velocity is no longer an indicator of signal velocity. Instead, a signal travels at the speed of the wavefront, which is c irrespective of the index of refraction.
The group velocity itself is usually a function of the wave's frequency. This results in group velocity dispersion (GVD), which causes a short pulse of light to spread in time as a result of different frequency components of the pulse travelling at different velocities. GVD is often quantified as the group delay dispersion parameter (again, this formula is for a uniform medium only):
  D  =            -              λ        c              ⁢                                        ⅆ            2                    ⁢          n                          ⅆ                      λ            2                              .      
If D is less than zero, the medium is said to have positive dispersion. If D is greater than zero, the medium has negative dispersion. If a light pulse is propagated through a normally dispersive medium, the result is the higher frequency components travel slower than the lower frequency components. The pulse therefore becomes positively chirped, or up-chirped, increasing in frequency with time. Conversely, if a pulse travels through an anomalously dispersive medium, high frequency components travel faster than the lower ones, and the pulse becomes negatively chirped, or down-chirped, decreasing in frequency with time.
The result of GVD, whether negative or positive, is ultimately temporal spreading of the pulse. Equivalently, GVD constitutes a linear channel whose frequency response can be closely approximated as
                    H        CD            ⁡              (        w        )              =          exp      ⁡              (                              -            j                    ⁢                                    D              ⁢                                                          ⁢                              λ                2                                                    4              ⁢                                                          ⁢              π              ⁢                                                          ⁢              c                                ⁢                                    L              ⁡                              (                                  w                  -                                      w                    s                                                  )                                      2                          )              ,where
w represents the angular frequency, ws is the center frequency of the band of interest, c is the speed of light, and L is the length of the fiber travelled by the communication signal. Note that in time domain, the above channel response has significant length, implying that a single impulse entering the fiber is dispersed broadly over time.
This makes dispersion management extremely important in optical communications systems based on optical fiber, since if dispersion is too high, a group of pulses representing a bit-stream will spread in time and merge together, rendering the bit-stream unintelligible. This phenomenon limits the length of fiber that a signal can be sent down without regeneration. One possible answer to this problem is to send signals down the optical fiber at a wavelength where the GVD is zero (e.g., around 1.3-1.5 μm in silica fibers), so pulses at this wavelength suffer minimal spreading from dispersion—in practice, however, this approach causes more problems than it solves because zero GVD unacceptably amplifies other nonlinear effects (such as four wave mixing). Instead, the solution that is currently used in practice is to perform dispersion compensation, typically by matching the fiber with another fiber of opposite-sign dispersion so that the dispersion effects cancel; such compensation is ultimately limited by nonlinear effects such as self-phase modulation, which interact with dispersion to make it very difficult to undo. Note that this method of compensation is performed optically, by using dispersion compensating fiber.
A third method is to perform dispersion compensation in the electronic domain. Since the dispersion is equivalently represented by a filter, it can be compensated by filtering at the transmitter or receiver. In order to enable such electronic dispersion compensation, communication is coherent, i.e., the optical front end preserves magnitude and phase information, which are both used by the electronics to perform dispersion compensation filtering.
In addition to chromatic dispersion, optical fibers also exhibit model dispersion caused by a waveguide (i.e. optical fiber) having multiple modes at a given frequency, each with a different speed. A special case of this is polarization mode dispersion (PMD), which comes from a superposition of two modes that travel at different speeds due to random imperfections that break the symmetry of the waveguide.
When a broad range of frequencies (a broad bandwidth) is present in a single wavepacket, such as in an ultrashort pulse or a chirped pulse or other forms of spread spectrum transmission, it may not be accurate to approximate the dispersion by a constant over the entire bandwidth, and more complex calculations are required to compute effects such as pulse spreading.
In particular, the dispersion parameter D defined above is obtained from only one derivative of the group velocity. Higher derivatives are known as higher-order dispersion. These terms are simply a Taylor series expansion of the dispersion relation β(ω) of the medium or waveguide around some particular frequency. Their effects can be computed via numerical evaluation of Fourier transforms of the waveform, via integration of higher-order slowly varying envelope approximations, by a split-step method (which can use the exact dispersion relation rather than a Taylor series), or by direct simulation of the full Maxwell's equations rather than an approximate envelope equation.
Coherent receivers perform both chromatic dispersion compensation using digital signal processing (DSP) of the analog-to-digital (ADC) output. Since long haul systems have large chromatic dispersion, the processing is complex and power-hungry.
It would be advantageous if pre-compensation could be performed in a transmitter, to minimize the effects of chromatic dispersion in signals received via an optical channel.